Assume That Adults Have Iq Scores That Are Normally Distributed,
This text discusses a z-score calculator and its uses, such as determining standard scores, converting between z-scores and probabilities, and finding the probability between two z-scores. The calculator also allows for step-by-step calculations and real-world applications. It explains the concept of the z-score, which follows a normal distribution with a mean of 0 and a standard deviation of 1. It includes a table to look up z-scores for specific percentiles and provides an example of finding the z-score for a given cumulative probability. Finally, it demonstrates how to use the calculator to find the raw-score based on a given z-score, mean, and standard deviation.
The probability that a randomly selected individual has an IQ between 75 and 117, given a mean of 107 and a standard deviation of 15 in a normally distributed IQ scores, can be calculated using the z-score formula and the standard normal distribution table.
First, we calculate the z-scores for both IQ values: For IQ = 75: [ z_1 = \frac{75 - 107}{15} = \frac{-32}{15} = -2.133 ]
For IQ = 117: [ z_2 = \frac{117 - 107}{15} = \frac{10}{15} = 0.667 ]
Using the standard normal distribution table, we find the corresponding probabilities:
- For (z = -2.133), the probability is approximately 0.0164.
- For (z = 0.667), the probability is approximately 0.7486.
Next, we find the probability between the two z-scores: [ P(75 < X < 117) = P(X < 117) - P(X < 75) ] [ P(75 < X < 117) = 0.7486 - 0.0164 = 0.7322 ]
Therefore, the closest value to the probability that a randomly selected individual has an IQ between 75 and 117 is 0.7320.
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